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I was reading the Linear Algebra Done Right textbook by Axler, and an extension for 5.27 has been troubling me.
Given linear operator $T \in \mathcal{L}(V)$ with $\lambda$ as an eigenvalue, we define $U = \mathrm{range\;}(T - \lambda I)$. Does there exist a $\lambda$-eigenvector $v \in U$? In other words, does there... |
Any Eigenvectors of $\lambda$, in Range of $T-\lambda I$? |
Any Eigenvectors of $\lambda$ in $\mathrm{range\;} T-\lambda I$? |
I was reading the Linear Algebra Done Right textbook by Axler, and an extension stemming from 5.27's proof 1 has been troubling me.
Given linear operator $T \in \mathcal{L}(V)$ with $\lambda$ as an eigenvalue, we define $U = \mathrm{range\;}(T - \lambda I)$. Does there exist a $\lambda$-eigenvector $v \in U$? In oth... |
Lets try to use index notation:
$$
y_i = {A^p}_i x_p
$$
So, for example,
$$
\begin{bmatrix}
y_1 \\ y_2
\end{bmatrix}
=
\begin{bmatrix}
{A^1}_1 & {A^2}_1 \\
{A^1}_2 & {A^2}_2
\end{bmatrix}
\begin{bmatrix}
x_1 \\ x_2
\end{bmatrix}
$$
Now we write the differential,
$$
dy_i = d{A^p}_i x_p = \delta^p_{p... |
I was reading the Linear Algebra Done Right textbook by Axler, and an extension stemming from 5.27's proof 1 has been troubling me.
Given linear operator $T \in \mathcal{L}(V)$ with $\lambda$ as an eigenvalue, we define $U = \mathrm{range\;}(T - \lambda I)$. Does there exist a $\lambda$-eigenvector $v \in U$? In oth... |
I do not understand one line in the proof of the following theorem. Within the proof of d) it's stated that, "if $y \in \mathfrak{h}$, then y commutes with x and hence also with s and n." I do not get why the fact that y commutes with x=s+n already implies that y commutes with both s and n. Does anybody know how the to... |
The conjecture is that the number $N(p)$ of different remainders modulo $p$ among these factorials $1!,\ldots (p-1)!$, satisfies
$$
\lim_{p\to \infty} \frac{N(p)}{p}=1-\frac{1}{e}\sim 0.6321205588285576784
$$
So $N(p)\ge \frac{p-1}{2}$ for $p$ large enough. For $p\le 10^7$ this bound is true by a computer verifica... |
I am trying to count all nilpotent matrices in $M_n(\mathbb R)$ upto similarity.
I did the same exercise for idempotent matrices and it was quite simple. I realised that rank of an idempotent matrix is a positive integer and two idempotent matrices are similar iff they have the same rank. So I got the answer $n+1$.... |
My question is very simple: A type $(m,n)$ tensor is an element of $V^{\otimes m}\otimes (V^*)^{\otimes n}$. Is there a reason/motivation, beyond more general definitions, to consider the dual space of $V$ in this definition? Or is it just convention?
I don't expect some mindblowing answer, so to speak, but maybe a ... |
why is a measure distribution of order 0? Question is from distribution theory and fourier transform.
Let $\Omega$ be an open domain in $\mathbb{R}^{n}$. A distribution $u$ in $\Omega$ is a linear functional on $C_{0}^{\infty}$ such that for every compact set $K \subset \Omega$ there are constants C and k such that $... |
Lets try to use index notation:
$$
y_i = {A^p}_i x_p
$$
So, for example,
$$
\begin{bmatrix}
y_1 \\ y_2
\end{bmatrix}
=
\begin{bmatrix}
{A^1}_1 & {A^2}_1 \\
{A^1}_2 & {A^2}_2
\end{bmatrix}
\begin{bmatrix}
x_1 \\ x_2
\end{bmatrix}
$$
Now we write the differential,
$$
dy_i = d{A^p}_i x_p = \delta^p_{p... |
I have to do the following exercise:
> Consider the endomorphism $h\colon\mathbb{R}^{6}\rightarrow\mathbb{R}^{6}$ with unique eigenvalues $0,\,2,\,8$. If $h$ is not diagonalizable and we know that dim$S(0)=1$, dim$S(2)\geq2$, dim$S(8)\geq2$, find all the possibilities for the elementary divisors and invariant factor... |
I have to do the following exercise:
> Consider the endomorphism $h\colon\mathbb{R}^{6}\rightarrow\mathbb{R}^{6}$ with unique eigenvalues $0,\,2,\,8$. If $h$ is not diagonalizable and we know that dim$S(0)=1$, dim$S(2)\geq2$, dim$S(8)\geq2$, find all the possibilities for the elementary divisors and invariant factor... |
Regarding your second question
> What can be the dual base of $C$?
You used the correct definition of dual basis
> We have $c_i^*(c_j)=\delta_{i,j}$
Hence the dual basis for $C$ would be $(c_1^*,c_2^*,c_3^*)$ where $\forall i=1,2,3$, $c_i^*$ of a polynomial gives the coefficient of the degree $i$ term.
R... |
Let $E = R_2[X]$.\
Let $a \in \mathbb{R}$.\
Let $ev_a : E \rightarrow\mathbb{R}, P \mapsto P(a)$.\
Let $P = a + bX + cX^2$.
Show that $B = (ev_1, ev_2, ev_3)$ is a base of $E^*$.
---
We have:
- $ev_1(P) = P(1) = a + b + c$
- $ev_2(P) = P(2) = a + 2b + 4c$
- $ev_3(P) = P(3) = a + 3b + 9c$
So let $A = \... |
I am trying to count all nilpotent matrices in $M_n(\mathbb R)$ upto similarity.
I did the same exercise for idempotent matrices and it was quite simple. I realised that rank of an idempotent matrix is a positive integer and two idempotent matrices are similar iff they have the same rank. So I got the answer $n+1$.... |
Let $E = R_2[X]$.\
Let $a \in \mathbb{R}$.\
Let $ev_a : E \rightarrow\mathbb{R}, P \mapsto P(a)$.\
Let $P = a + bX + cX^2$.
Show that $B = (ev_1, ev_2, ev_3)$ is a base of $E^*$.
---
**Proof** :
We have:
- $ev_1(P) = P(1) = a + b + c$
- $ev_2(P) = P(2) = a + 2b + 4c$
- $ev_3(P) = P(3) = a + 3b + 9c$
... |
I was reading the Linear Algebra Done Right textbook by Axler, and an extension stemming from 5.27's proof 1 has been troubling me.
Given linear operator $T \in \mathcal{L}(V)$ with $\lambda$ as an eigenvalue, we define $U = \mathrm{range\;}(T - \lambda I)$. Does there exist a $\lambda$-eigenvector $v \in U$? In oth... |
I was reading the Linear Algebra Done Right textbook by Axler, and an extension stemming from 5.27's proof 1 has been troubling me. This is a closer look into the book’s construction of $U$, an invariant subspace. I wanna find out if:
Given linear operator $T \in \mathcal{L}(V)$ with $\lambda$ as an eigenvalue, we d... |
These vertices are in two circles. Note that we must take $11$ vertices from one circle and $12$ from the other. Otherwise we take $13$ or more vertices from one circle and since $2\cdot13>25$ there would be adjacent chosen vertices.
Suppose we take $12$ vertices from the inner circle. There are $25$ ways to do it.... |
Lets try to use index notation:
$$
y_i = {A^p}_i x_p
$$
So, for example,
$$
\begin{bmatrix}
y_1 \\ y_2
\end{bmatrix}
=
\begin{bmatrix}
{A^1}_1 & {A^2}_1 \\
{A^1}_2 & {A^2}_2
\end{bmatrix}
\begin{bmatrix}
x_1 \\ x_2
\end{bmatrix}
$$
Now we write the differential,
$$
dy_i = d{A^p}_i x_p = \delta^p_{p... |
>If $A$ is a skew-symmetric Matrix then prove that $I-A$ is non singular, where $I$ is the identity Matrix of the order equal to that of $A$.
I know that this question can be done by using the concept of eigenvalues. But I want a solution that does not involve that. I have one solution but it is like very difficult ... |
Let $k$ denote the field of real numbers, or the field of complex numbers. Given any vector space $V$ over $k$, it is always possible define a metric on $V$ (say the [discrete metric][1]), or a norm on $V$ ([by choosing a Hamel basis][2]). However, if the norm you define has no relation to the metric, then there is no ... |
In Karl Rubin's book Euler System, he gives a definition of selmer groups as below:[![enter image description here][1]][1]
Then he states as in lemma 5.3 on page 12 that the selmer group with upper tag is equal to a first cohomology group with appropiate assumption on A:
[![enter image description here][2]][2]
H... |
I proved a proposition in $\mathbb{R}^2$ using a cartersian coordinate system. Now I want to provide another proof of the same proposition without the cartesian coordinate system. Anyway I need to express some half-planes through the scalar product. My skill in geometry are very basic and I'm having some difficulties. ... |
Let $f$ and $g$ be functions such that for all real numbers $x$ and $y$ such that:
**$g(f(x + y)) = f(x) + (x+y)g(y)$.**
The value of $(0) + (1)+. . . (2024)$ is?
I found the question on Mathematics Stack Exchange:
https://math.stackexchange.com/questions/4863341/finding-the-value-of-functions?noredire... |
How can different representations of the same integer be equivalent? |
Setting: Let $Y$ be a metric space and let $a_{n,k}\in Y$ for all $n\in\mathbb{N}$ and $k\in\mathbb{N}$. Suppose $a_{n,k}\to a_{\bullet, k}$ uniforly as $n\to\infty$. Suppose the sequences $\{a_{n,k}\}\to A_n$ as $k\to \infty$ for each $n\in\mathbb{N}$, .
---
There are two questions in my mind:
1. Is it pos... |
Are conergenet sequences closed under uniform convergence? |
One of the properties of primes is that
>Suppose a prime $p\mid ab$ then $p\mid a$ or $p\mid b$
This follows from the fundamental theorem of arithmetic.
Now since $7\mid b\cdot b$ then either $7\mid b$ or $7\mid b$. So $7\mid b$.
Notice that you have only proven that if $7\mid a$ then $7\mid b$. The case s... |
Good morning,
I need help with the following integral
$$ \int_0 ^1 \left(\log (y) \frac{y^{a -1} {}_2F_1(a ,a ;2 a ;y)}{y-x} - \log (x) \frac{x^2 y^{a -3} {}_2F_1(a ,a ;2 a;y)}{y-x}\right) dy$$
where ${}_2F_1(a ,a ;2 a ;y)$ is the Gauss hypergeometric function, $a>0$ and $0<x<1$. I am looking for a result in terms... |
> Suppose that $$x + y + z = 203.$$ What would be the maximum value of $$x + (xy)^{1/2} + (xyz)^{1/3}?$$
The solution is to assume that
$$\frac43\cdot (x+y+z)\ge x+(xy)^{1/2}+(xyz)^{1/3}$$ and prove it with AM-GM inequality. I was just wondering how would one come up with that assumption normally? And if there's... |
I am reading Silverman's *The Arithmetic of Elliptic Curves*. I have a question regarding Example 3.5, which is the following:
>[![enter image description here][1]][1]
I have doubts about the equality $[-Y^2,Y(X-Z))]=[-Y,X-Z]$. Two points $[x_0,\ldots, x_n]$,$[x_0',\ldots, x_n']$ in the projective space are the sam... |
Suppose $(x_n)_{n=1}^{k}\ $ is a decreasing $k-$tuple of positive real numbers. Let $(y_n)_{n=1}^{2k}\ $ be two copies of $(x_n)_{n=1}^{k},\ $ that is, a sequence containing $x_n$ twice for each $n\in\{1,2,\ldots,k\}.$ Does there exist a permutation $(\sigma(y_n))_{n=1}^{2k}\ $ of $(y_n)_{n=1}^{2k}\ $ and an increasing... |
Using two copies of a decreasing $k-$tuple, can you form a convex decreasing $k-$tuple greater than the original sequence? |
(W(t), t >= 0) is Brownian motion.
Let 0<s<t. What is the distribution of 2W(s)-W(t)?
If s>t, then 2W(s)-W(t) ~ W(4s)-W(t) ~ N(0, 4s-t), but here we have t>s. Does this calculations still hold? |
My attempt at it has been as follows:
first we take the case for all objects distinct and select k objects from the n objects in nCk ways, then we take the case of 2 objects being equal, and since these 2 would need to be consecutive there are n-1 such cases and the ways to select objects would be nCk-1 since we nee... |
I'm taking Non-Linear PDEs course this semester. I'm stuck in this HW problem.
Solve the IVP:
$ u_t+uu_x =0, \ \ x \in \mathbb{R}, \ t\ge 0 $
$$
u(x,0) = \left\lbrace
\begin{aligned}
& 1 &&\text{for}\; x < -1\\
& -x &&\text{for}\; {-1}\leqslant x \leqslant 0 \\
& 1 &&\text{for}\; x > 0\, .
\end{alig... |
Suppose $(x_n)_{n=1}^{k}\ $ is a decreasing $k-$tuple of positive real numbers. Let $(y_n)_{n=1}^{2k}\ $ be two copies of $(x_n)_{n=1}^{k},\ $ that is, a sequence containing $x_n$ twice for each $n\in\{1,2,\ldots,k\}.$ Does there exist a permutation $(\sigma(y_n))_{n=1}^{2k}\ $ of $(y_n)_{n=1}^{2k}\ $ and an increasing... |
Are convergent sequences closed under uniform convergence? |
I proved a proposition in $\mathbb{R}^2$ using a cartersian coordinate system. Now I want to provide another proof of the same proposition without the cartesian coordinate system. Anyway I need to express some half-planes through the scalar product. My skill in geometry are very basic and I'm having some difficulties. ... |
Let $f$ and $g$ be functions such that for all real numbers $x$ and $y$ such that:
**$g(f(x + y)) = f(x) + (x+y)g(y)$.**
The value of $(0) + (1)+. . . (2024)$ is?
I found the question on Mathematics Stack Exchange:
https://math.stackexchange.com/questions/4863341/finding-the-value-of-functions?noredire... |
Is the only continuous outer automorphism of $\operatorname{GL}(n, \mathbb{R})$ the transpose inverse map $g \mapsto (g^\intercal)^{-1}$? If not, what other continuous outer automorphisms are there? |
What are the continuous outer automorphisms of the general linear group? |
>Let $A_1, A_2, A_3, ..., A_n (n>1)$ be independent events such that $P(A_i)=\frac{1}{i}$. What is the smallest possible size of $\Omega$? And what if $P(A_i)=\frac{1}{e^i}$?
I imagine this as tossing $n$ unfair coins, where the first one always lands heads and the others can land either heads or tails, hence ther... |
Let $f$ and $g$ be functions such that for all real numbers $x$ and $y$:
**$g(f(x + y)) = f(x) + (x+y)g(y)$.**
The value of $(0) + (1)+. . . (2024)$ is?
I found the question on Mathematics Stack Exchange:
https://math.stackexchange.com/questions/4863341/finding-the-value-of-functions?noredirect=1#comme... |
$g(f(x + y)) = f(x) + (x+y)g(y).$ Value of $(0) + (1)+\dots+ (2024)$? |
I have the center of a circle $\vec{c}$ in 3 space and the radius $r$. I also have a unit vector $\hat{v}$ defining the orientation of the plane of the circle.
I wish to parameterize this circle and place a single point on the circle.
My logic is, imagine I have a point in polar coordinates.
I could describe... |
Let $(W(t))_{t\geq 0}$ be a Brownian motion and $0<s<t.$
What is the distribution of $2W(s)-W(t)$?
If $s>t$, then $2W(s)-W(t) \sim W(4s)-W(t) \sim N(0, 4s-t)$, but here we have $t>s$.
Does this calculations still hold? |
Let $f$ and $g$ be functions such that for all real numbers $x$ and $y$:
**$g(f(x + y)) = f(x) + (x+y)g(y)$.**
The value of $(0) + (1)+. . . (2024)$ is?
I found the question on Mathematics Stack Exchange:
https://math.stackexchange.com/questions/4863341/finding-the-value-of-functions?noredirect=1#comme... |
Jacobi and Madden found infinitely many primitive solutions to,
$$a^4+b^4+c^4+d^4 = (a+b+c+d)^4$$
using an elliptic curve. We will use a different approach that, like the method for,
$$a^4+b^4+c^4 = d^4$$
discussed in [this post][1], also yields **pairs** of solutions.
---
**I. The system**
Define,... |
How to define the smallest exponent k for that $n^{k} > n!$ ?
Does someone know whether exists a formula for that?
For example:
for $n = 2, k = 2,$ since $2^{2} > 2!$
for $n = 3, k = 2,$ since $3^{2} > 3!$
for $n = 4, k = 3,$ since $4^{3} > 4!$
Thanks in advance |
First, you have to answer the question : what is the mathematical definition of an half-plane? Or even what is a half-space? Generalizing, it won't be much harder to answer.
Let $E$ be a real vector space (for example, $\mathbb R^2, \mathbb R^3, \mathbb R^4,...)$ with its Euclidean structure, where the dot product i... |
I have the center of a circle $\vec{c}$ in 3 space and the radius $r$. I also have a unit vector $\hat{v}$ defining the orientation of the plane of the circle.
I wish to parameterize this circle and place a single point on the circle.
My logic is, imagine I have a point in polar coordinates.
I could describe... |
Let $f$ and $g$ be functions such that for all real numbers $x$ and $y$:
$$g(f(x + y)) = f(x) + (x+y)g(y).$$
The value of $(0) + (1)+\ldots (2024)$ is?
I found the question on Mathematics Stack Exchange:
https://math.stackexchange.com/questions/4863341/finding-the-value-of-functions?noredirect=1#commen... |
# Background
I am trying to understand the paper ["On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg][1]. However, despite my efforts to decipher the paper, I can't follow what it is describing. This paper is written around complex differential geometry and topology, and appear... |
Can you explain how to project this vector onto the complex plane to someone who hasn't learned Differential Geometry or Topology? |
First, you have to answer the question : what is the mathematical definition of an half-plane? Or even what is a half-space? Generalizing, it won't be much harder to answer.
Let $E$ be a real vector space (for example, $\mathbb R^2, \mathbb R^3, \mathbb R^4,...)$ with its Euclidean structure, where the dot product i... |
I was trying to prove it, however am pretty confident that my argument is incorrect, as I don't even use the fact of completeness of the metric space. Can anybody help me to figure out where I am wrong with my attempt? Here is it:
In a complete metric space, I need to prove that total boundedness implies compactness... |
I have to do the following exercise:
> Consider the endomorphism $h\colon\mathbb{R}^{6}\rightarrow\mathbb{R}^{6}$ with unique eigenvalues $0,\,2,\,8$. If $h$ is not diagonalizable and we know that dim$S(0)=1$, dim$S(2)\geq2$, dim$S(8)\geq2$, find all the possibilities for the elementary divisors and invariant factor... |
I'm reading [Introduction to Mathematical logic][1] by Sam Buss. I'm currently going through propositional logic and I can't understand why (c) is correct:
> Theorem I.13. Suppose that Γ and ∆ are sets of formulas, and Γ ⊆ ∆.
>
> (a) If φ satisfies ∆, then φ satisfies Γ.
>
> (b) If ∆ is satisfiable, then Γ is... |
Suppose there was a predicate, in the language of 1st order $ \mathsf {PA} $, such that it is only true for standard natural numbers i.e. it accepts ALL and ONLY standard natural number, and it **rejects** *any* and *every* non-standard natural number.
Since PA is $ \boldsymbol \omega $-consistent, therefore any su... |
Suppose there was a predicate, in the language of 1st order $ \mathsf {PA} $, such that it is only true for standard natural numbers i.e. it accepts ALL and ONLY standard natural number, and it **rejects** *any* and *every* non-standard natural number.
Since PA is $ \boldsymbol \omega $-consistent, therefore any su... |
I am trying to count all nilpotent matrices in $M_n(\mathbb R)$ upto similarity.
I did the same exercise for idempotent matrices and it was quite simple. I realised that rank of an idempotent matrix is a non-positive integer and two idempotent matrices are similar iff they have the same rank. So I got the answer $n... |
I am studying *Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan*. The following is the beginning of Appendix D:
[![enter image description here][1]][1]
I could not understand the last sentence (the one contains D.2.). It's like a theorem that is well known in Fourier Analy... |
I just found this kind of “conjecture” recently. Define the set of positive prime numbers $P$. Suppose $$ n \in P, n\geq 3. $$ Define $S := \{1,p_1,\dots,p_{x-1}\}$ that are primes and $1$ which are smaller than $n$.
(in which $p_{x-1}$ is the prime directly before n)
1. Can you find at most 2 distinct $a, b \in S$ (... |
Setting: Let $Y$ be a metric space and let $a_{n,k}\in Y$ for all $n\in\mathbb{N}$ and $k\in\mathbb{N}$. Suppose $a_{n,k}\to a_{\bullet, k}$ uniforly as $n\to\infty$. Suppose the sequences $\{a_{n,k}\}\to A_n$ as $k\to \infty$ for each $n\in\mathbb{N}$, .
---
There are two questions in my mind:
1. Is it pos... |
First, you have to answer the question : what is the mathematical definition of an half-plane? Or even what is a half-space? Generalizing, it won't be much harder to answer.
Let $E$ be a real vector space (for example, $\mathbb R^2, \mathbb R^3, \mathbb R^4,...)$ . Then let $f\neq0$ be a linear form on $E$. And let ... |
There is an equilateral triangle with one cafe at each vertice. They are called R, S, and M.
I tried to solve this problem but got stuck. Here is my work:
Since there are six cafes left to visit, knowing that Cafes R and M are the start and end cafes, I have two choices for each cafe. Hence, the answer is 2^6 or ... |
Let $k$ denote the field of real numbers, or the field of complex numbers. Given any vector space $V$ over $k$, it is always possible define a metric on $V$ (say the [discrete metric][1]), or a norm on $V$ ([by choosing a Hamel basis][2]). However, if the norm you define has no relation to the metric, then there is no ... |
$p\implies q$ does not neccisarily indicate that $q\implies p$. Which is why there exists a double-sided implication: $\iff$.
$p\iff q$ is the same as saying $(p\implies q)\land(q\implies p)$. However, sometimes when I am illustrating my working out, I use the one-sided $\implies$ symbol even when using the $\iff$ s... |
Does $(p\implies q)\implies\neg( q\implies p)$? |
Can you help me to confirm if this is the right approach for determine a valid kernel function?
$ k(x, x') = \tanh(3x^Tx' - 8) $ a valid kernel function?
To determine if this function is a valid kernel, we must show that it corresponds to an inner product in some feature space.
My attempt:
Let's consider th... |
There is an equilateral triangle with one cafe at each vertice. They are called R, S, and M.
I tried to solve this problem but got stuck. Here is my work:
After knowing that R and M are the endpoints, we can alternate to the 3 cafes for 5 steps or 6 times. I have two choices for each cafe. Hence, the answer is 2^... |
>Find the best possible upper bound on $P(X \geq 100)$ if $g_X(10)=10$ ($g_X$ is a generating function).
I started by writing the equation to find possible generating function $g_X(10)=(1-p)\cdot10^0+p\cdot10^{100} = 10$.
Its solution is $p = \frac{9}{10^{100}-1}$, hence $P(X\geq 100) = \frac{9}{10^{100}-1}$.
... |
Suppose $(x_n)_{n=1}^{k}\ $ is a decreasing $k-$tuple of positive real numbers. Let $(y_n)_{n=1}^{2k}\ $ be two copies of $(x_n)_{n=1}^{k},\ $ that is, a sequence containing $x_n$ twice for each $n\in\{1,2,\ldots,k\}.$ Does there exist a permutation $(\sigma(y_n))_{n=1}^{2k}\ $ of $(y_n)_{n=1}^{2k}\ $ and an increasing... |
I was able to find a demonstration,
starting from the right part, <br/>
I wanted to start from the left part, in order to check if I understood well my knowdlege,
Starting from left or right does make any difference in how it's complex the resolution? And, in case yes, what criteria should I use to identify which ... |
Boole Algebra, ab+bc+ca=(a+b)(b+c)(c+a), how to solve starting from left? |
My question is, I hope, simple. I have a smooth, properly embedded one dimensional submanifold $M \subset R^n, \; n > 2$ (i.e. a smooth curve, which is actually the level set of a smooth function $\phi : R^n \rightarrow R^{n-1}$). I know from separate considerations that the projection $\pi$ of $M$ onto its first two c... |
Is the image of the projection of a smooth curve onto a subset of its coordinates a manifold if the projection is an injective immersion? |
I was able to find a demonstration,
starting from the right part, <br/>
I wanted to start from the left part, in order to check if I understood well my knowdlege,
Starting from left or right does make any difference in how it's complex the resolution? And, in case yes, what criteria should I use to identify which ... |
Let $k$ denote the field of real numbers, or the field of complex numbers. Given any vector space $V$ over $k$, it is always possible define a metric on $V$ (say the [discrete metric][1]), or a norm on $V$ ([by choosing a Hamel basis][2]). However, if the norm you define has no relation to the metric you define, then t... |
Let $k$ denote the field of real numbers, or the field of complex numbers. Given a vector space $V$ over $k$, it is always possible define a metric on $V$ (say the [discrete metric][1]), or a norm on $V$ ([by choosing a Hamel basis][2]). However, if the norm you define has no relation to the metric you define, then the... |
I don't have an answer to your question, but I did search through quite a few set theory books this morning and I made notes of what I found in case you or others are interested.
The topic seems less covered in books than I expected, and I suspect you'll have to consult journal articles to find much of significance ... |
For each $n$ define the sequence $y_n= \int_{0}^\infty (1-e^{-\frac{x^2}{4}})^n e^{-x}dx$. By dominated convergence theorem, it is clear that $y_n$ converges to $0$. I am interested in finding the rate at which $y_n$ decays. Mainly I am interested to know if the decay is exponential in $n$ or not i.e $y_n$ is roughly $... |
I am trying to construct the character table for the symmetry group of Cube, $O_h$, which as 48 elements. I figured out that there are 10 classes. Now, as a consequence of the great orthogonality theorem, I have $\sum l_i^2 = 48$ ...(1), and the number of irreducible representations is equal to the number of classes. S... |
I am trying to count all nilpotent matrices in $M_n(\mathbb R)$ up to similarity.
I did the same exercise for idempotent matrices and it was quite simple. I realised that rank of an idempotent matrix is a non-negative integer and two idempotent matrices are similar iff they have the same rank. So I got the answer $... |
In [this thread about the derivation of lagrange multipliers][1] as well as [the wikipedia article][2] the Lagrangian is defined as
$ F(\vec{x}, \vec{\lambda}) = f(\vec{x}) - \langle \vec{\lambda}, \vec{g(x)} \rangle $.
I thought that $\lambda$ had to be a scalar in order to constrain $ \nabla g $ and $ \nabla f... |
Can lagrange multipliers be vectors? |
>Find the best possible upper bound on $P(X \geq 100)$ if $g_X(10)=10$ ($g_X$ is a generating function), where $X$ is a discrete random variable.
I started by writing the equation to find possible generating function $g_X(10)=(1-p)\cdot10^0+p\cdot10^{100} = 10$.
Its solution is $p = \frac{9}{10^{100}-1}$, hence $... |
I am seeking a motivation for the definition of the curvature of a plane curve. How did people come with the idea of the definition of the curvature? Below I am more specific.
The fundamental theorem for plane curves states the following. Giving two curves $\alpha$ and $\beta$ there exists a rigid motion $M:\mathbb ... |
To answer your question: when can $(a-3)(a-7)$ be a perfect square?
Suppose
$$(a-3)(a-7)=k^2, $$
for some integer $k$. Then
$$a^2-10a+21-k^2=0.$$
This implies that
$$a = \dfrac{10\pm\sqrt{100-4(21-k^2)}}{2},$$
implying that
$$a = \dfrac{10\pm 2\sqrt{4+k^2}}{2}.$$
As $a$ has to be an integer, we look for... |
>Is $\mathbb{R}^3\setminus\{(x,y,z)\in\mathbb{R}^3\mid x\leq 1,y\leq 1,-1\leq z\leq 1\}$ a star domain.
**My approach:**
Let be $N:=\{(x,y,z)\in\mathbb{R}^3\mid x\leq 1,y\leq 1,-1\leq z\leq 1\}$ and $M:=\mathbb{R}^3\setminus N$ and $a\in M$ such that $a_3>1$ and $b=(0,0,1)\in N$.
We define $S_{a,b}:\mathbb{... |
Is $\mathbb{R}^3\setminus\{(x,y,z)\in\mathbb{R}^3\mid x\leq 1,y\leq 1,-1\leq z\leq 1\}$ a star domain? |
I'll take a crack at this.
Any time you make a logical argument, you maybe presume some things that are true (your *axioms*) and always presume some rules about how you can work with true premises to reach true conclusions (your *rules*). Together, they form a *theory*.
Maybe I'm doing some set theory, and I'm wo... |
As we know for $s, t >0,$ we have
$s f(x) + t f(y) = (s + t) f(z)$, for continues function $f$ on $[a, b]$ ($x, y\in (a, b)$).
Is it true for $s f(x) - t f(y) = (s - t) f(z)$ ???? |
I am trying to prepare for next semester in my university, so I tried to solve some of the questions with complex numbers. So the main question is, how to factor this task (find modulus, real and imaginary part)?
$$z = \frac{(1+i)(1+i)^2(i+1)^3...(1+i)^{20}}{i^0+i^2+i^4+i^6+...+i^{20}}$$ |
I am trying to prepare for next semester in my university, so I tried to solve some of the questions with complex numbers. So the main question is, how to solve this task (find modulus, real and imaginary part)?
$$z = \frac{(1+i)(1+i)^2(i+1)^3...(1+i)^{20}}{i^0+i^2+i^4+i^6+...+i^{20}}$$ |
>Is $\mathbb{R}^3\setminus\{(x,y,z)\in\mathbb{R}^3\mid x\leq1,y\leq1,-1\leq z\leq1\}$ a star domain.
**My approach:**
Let be $N:=\{(x,y,z)\in\mathbb{R}^3\mid x\leq1,y\leq 1,-1\leq z \leq1\}$ and $M:=\mathbb{R}^3\setminus N$ and $a\in M$ such that $a_3>1$ and $b=(0,0,1)\in N$.
We define $S_{a,b}:\mathbb{R}\t... |
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